तृतीयासप्तम्योर्बहुलम्

Operational rule vidhi

The change to अम् of the Instrumental -3rd case and Locative -7th-case-affixes coming after an अव्ययीभाव compound that ends in अ , occurs diversely.

Word by word (padaccheda)

Target तृतीया- सप्तम्योः ṣaṣṭhī · dvivacanaSubstitute / term बहुलम् prathamā · ekavacana

Target Ṣaṣṭhī (6th case): names what gets replaced, "in place of X" (1.1.49 षष्ठी स्थानेयोगा).
Substitute / term Prathamā (1st case): in operational rules, the substitute or the thing introduced; in definitions, the term or its referent.

Dotted words are technical terms, dashed ones pratyāhāras (names for classes of sounds or affixes). Hover or tap one to see its card.

Full reading

तृतीया-सप्तम्योः बहुलम्

Pāṇini leaves out words that carry over from earlier sūtras. Expand to see the rule as a reader actually understands it.

As a rewrite rule (sketch from case endings)

तृतीया- सप्तम्योः → बहुलम्

Replace tṛtīyā-saptamyoḥ with bahulam. Inherited words (anuvṛtti) may add further conditions.

Commentary

The change to अम् of the Instrumental -3rd case and Locative -7th-case-affixes coming after an अव्ययीभाव compound that ends in अ , occurs diversely.
उपकुम्भेन or उपकुम्भम् कृतं 'done by upakumbha'; उपकुम्भे निधेहि or उपकुम्भम् निधेहि 'put it in the upakumbha.'

Vart:- The अम् substitution is invariable and not optional in the locative case when the Avyayibhava compound denotes prosperity (II.1.6.) or a compound of rivers 2.1.20 and 2.1.21 or a compound having a numeral for its member 2.1.19; as, सुमद्रम् 'well or prosperous with the Madras'; सुमगधम् 'well or prosperous with the Magadhas'; similarly उन्मत्तगङ्गम् । लोहितगङ्गम् । एकविंशतिभारद्वाजम् । The word 'diversely' establishes all these even without the Vartika.

Śrīśa Chandra Vasu, The Ashtadhyayi of Panini (1897), via ashtadhyayi.com.